Question

If p, q and 23 is an increasing arithmetic sequence and p and q are prime numbers, then $p+q=$

• A. 22
• B. 24
• C. 26
• D. 28
• E. 30

Hint:

Number Theory Tip: When equations involve primes, checking for odd/even parity is the fastest way to restrict the possible values and eliminate guesswork!

Answer 1

The Correct Option is D

Concept:
In an Arithmetic Progression (A.P.), the difference between consecutive terms is constant. For three terms $a, b, c$ in an A.P., the middle term is the arithmetic mean of the other two, meaning $2b = a + c$.

Step 1: Set up the A.P. equation.
Given that $p, q, 23$ form an increasing arithmetic sequence, apply the arithmetic mean property: $$2q = p + 23$$

Step 2: Determine the parity of p.
Since $2q$ must be an even number (as it's a multiple of 2), the sum $p + 23$ must also be even. Because $23$ is odd, $p$ must be an odd number. Therefore, $p$ cannot be the only even prime number, $2$.

Step 3: Test odd prime values for p.
Since the sequence is increasing, $p < q < 23$. Let's test primes for $p$: If $p = 3$: $2q = 3 + 23 = 26 \implies q = 13$ (Both are prime. Sequence: 3, 13, 23) If $p = 5$: $2q = 5 + 23 = 28 \implies q = 14$ (14 is not prime) If $p = 7$: $2q = 7 + 23 = 30 \implies q = 15$ (15 is not prime) If $p = 11$: $2q = 11 + 23 = 34 \implies q = 17$ (Both are prime. Sequence: 11, 17, 23)

Step 4: Calculate the possible sums.
We found two valid pairs of primes that work: Pair 1: $p = 3, q = 13 \implies p + q = 16$ Pair 2: $p = 11, q = 17 \implies p + q = 28$

Step 5: Match with the given options.
Checking the provided multiple-choice options, 16 is not listed, but 28 is. Therefore, the correct pair must be $p = 11$ and $q = 17$. Hence the correct answer is (D) 28.